Read The Nature of Mathematics Online

THE SCIENCE OF DYNAMICS 59 

drawn from the origin O, and the curve of velocities is a 
straight line parallel to the jr-axis. If the velocity is variable, 
the curve of spaces is never a straight line ; but if the motion 
is uniformly accelerated, the curve of velocities is a straight 
line like OB. The relations between the curve of spaces, the 
curve of velocities, and the areas of such curves AOB are, as 
we shall see, relations which are at once expressible by the 
"differential and integral calculus," indeed, it is mainly be- 
cause of this important illustration of the calculus that the 
elementary problems of dynamics have been treated here. 
And the measurement of velocity in the case where the velocity 
varies from time to time is an illustration of the formation of 
the fundamental conception of the differential calculus. 

It may be remarked that the finding of the velocity of a 
particle at a given instant and the finding of a tangent to a 
curve at a given point are both of them the same kind of 
problem the finding of the " differential quotient " of a func- 
tion. We will now enter into the matter more in detail. 

Consider a curve of spaces. If the motion is uniform, the 
number measuring any increment of the distance divided by 
the number measuring the corresponding increment of the time 
gives the same value for the measure of the velocity. But if 
we were to proceed like this where the velocity is variable, we 
should obtain widely differing values for the velocity. How- 
ever, the smaller the increment of the time, the more nearly 
does the bit of the curve of spaces which corresponds to this 
increment approach straightness, and hence uniformity of in- 
crease (or decrease) of s. Thus, if we denote the increment of 
t by " Az," where " A " does not stand for a number but for the 
phrase "the increment of," and the corresponding increment 
(or decrement) of s by " As," we may define the measure of 

As 
average velocity in this element of the motion as -r;. But, 

however small At is, the line represented by As is not, usually 
at least, quite straight, and the velocity at the instant t, which, 
in the language of Leibniz's differential calculus, is defined as 
the quotient of " infinitely small " increments and symbolised 

ds 

by , the A's being replaced by d's when we consider 
at 

"infinitesimals," appears to be only defined approximately.